There is free movement between countries in Europe. I suppose there are generally some language barriers, but to me the situation seems fundamentally the same.

But is there freedom of access to social systems?

https://ind.nl/en/consequences-for-right-of-residence-when-a...

Nope, the countries put up significant barriers to access social safety nets (permanent residency durations, proof of working during time before that, etc).

In the US you can generally establish residency in a different state in the amount of time it takes to get a USPS change of address notice and a billing statement of some sort showing that you’re able to receive mail there.

No, there is only a limited the freedom of travel between countries in EU (not Europe), but not of movement. For many years people from the newly admitted countries in EU had no right to work in the rest of EU. Most of the diplomas are not automatically valid in other countries and in some cases, not at all - example: pilot license for light planes, I have a Romanian license and I can fly in Greece a plane registered in Romania, but not a plane registered in Greece. If I switch to a Greek license, with training and exams, I am not allowed to fly Romanian planes :(

Very different systems, very different safety nets, and non-trivial language barriers.

Yeah you can move wherever but you're not eligible for benefits, and the hiring managers want to know you speak fluent $LANGUAGE and have the right certifications from whatever credentialing system exists there.

Common currency and regulatory standards, but it's not like you're just moving from Spain to Poland, at least not without serious hurdles.